A VARIATIONAL FORMULA FOR SOME RANDOM FIELDS: AN ANALOGUE OF ITO'S FORMULA

Abstract

We shall first establish a canonical representation of a Gaussian random field X(C) indexed by a smooth contour C in terms of two-dimensional parameter white noise. Then, we take a nonlinear function F(X(C)) of the X(C) and obtain its variation when C deforms slightly. The variational formula is analogous to the Ito formula for a stochastic process X(t), but somewhat simpler.